Integrand size = 21, antiderivative size = 440 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
Time = 1.62 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {45, 5355, 12, 6853, 6874, 759, 21, 733, 435, 972, 946, 174, 552, 551, 849, 858, 430} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d \left (1-c^2 x^2\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}} \]
[In]
[Out]
Rule 12
Rule 21
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 946
Rule 972
Rule 5355
Rule 6853
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {b \int \frac {2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c} \\ & = -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {(2 b) \int \frac {8 d^2+12 d e x+3 e^2 x^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^3} \\ & = -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {8 d^2+12 d e x+3 e^2 x^2}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {8 d^2}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {3 e^2 x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {12 b d \left (1-c^2 x^2\right )}{c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {e}{2}-\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {12 b d \left (1-c^2 x^2\right )}{c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b c d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b c d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b (c d-e) (c d+e) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (16 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (4 b (c d-e) (c d+e) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{c^2 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (32 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}} \\ & = \frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.07 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.95 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {a d^3 \left (1+\frac {e x}{d}\right )^{5/2} B_{-\frac {e x}{d}}\left (3,-\frac {3}{2}\right )}{e^3 (d+e x)^{5/2}}+\frac {b \left (-\frac {c^3 \left (e+\frac {d}{x}\right )^3 x^3 \left (\frac {4 c d \sqrt {1-\frac {1}{c^2 x^2}}}{3 e^2 \left (-c^2 d^2+e^2\right )}-\frac {16 \csc ^{-1}(c x)}{3 e^3}+\frac {2 \csc ^{-1}(c x)}{3 e \left (e+\frac {d}{x}\right )^2}+\frac {4 \left (-c d e \sqrt {1-\frac {1}{c^2 x^2}}-2 c^2 d^2 \csc ^{-1}(c x)+2 e^2 \csc ^{-1}(c x)\right )}{3 e^2 \left (-c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{5/2}}-\frac {2 \left (e+\frac {d}{x}\right )^{5/2} (c x)^{5/2} \left (\frac {2 \left (3 c^2 d^2 e-3 e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (8 c^3 d^3-9 c d e^2\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 c d e \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{3 (c d-e) e^3 (c d+e) (d+e x)^{5/2}}\right )}{c^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1025\) vs. \(2(401)=802\).
Time = 9.03 (sec) , antiderivative size = 1026, normalized size of antiderivative = 2.33
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1026\) |
default | \(\text {Expression too large to display}\) | \(1026\) |
parts | \(\text {Expression too large to display}\) | \(1041\) |
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
[In]
[Out]